The diagram shows the probability density function, $$\(\mathrm{f}(x)\)$$, of a random variable $$\(X\)$$. For $$\(0 \leqslant x \leqslant a\)$$, $$\(\mathrm{f}(x)=k ;\)$$ elsewhere $$\(\mathrm{f}(x)=0\)$$. Given that $$\(\operatorname{Var}(X)=3\)$$, find $$\(a\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_w20_qp_63 Year：2020 Question No：4(b)
Answer:
\((\) Mean \(=)\) their \(k \times \frac{a^{2}}{2}\left(=\frac{a}{2}\right)\)
\(\frac{1}{a} \int_{0}^{a} x^{2} \mathrm{~d} x\left(=\frac{a^{2}}{3}\right)\)
\(-\left(\frac{a^{\prime}}{2}\right)^{2}\left(=\frac{a^{2}}{12}\right)\)
\(\left(\frac{a^{2}}{12}=3\right) a=6\)
\(\frac{1}{a} \int_{0}^{a} x^{2} \mathrm{~d} x\left(=\frac{a^{2}}{3}\right)\)
\(-\left(\frac{a^{\prime}}{2}\right)^{2}\left(=\frac{a^{2}}{12}\right)\)
\(\left(\frac{a^{2}}{12}=3\right) a=6\)
Knowledge points:
6.3.2 use a probability density function to solve problems involving probabilities, and to calculate the mean and variance of a distribution. (Including location of the median or other percentiles of a distribution by direct consideration of an area using the density function.) (Explicit knowledge of the cumulative distribution function is not included.)
Solution:
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